Why are there no Cartesian products in grammar?

This post, I think, doesn’t rise above the level of “musings.” I think there’s something here, but I’m not sure if I can articulate it properly.

An adequate scientific theory is one in which facts about nature are reflected in facts about the theory. Every entity in the theory should have an analogue in nature, relations in the theory should be found in nature, and simple things in the theory should be ubiquitous in nature. This last concern is at the core of minimalist worries about movement—early theories saw movement as complex and had to explain its ubiquity, while later theories see it as simple and have to explain the constraints on it. But my concern here is not minimalist theories of syntax, but model-theoretic semantics.

Model theories of semantics often use set-theory as their formal systems,[1]Yes, I know that there are many other types of model theories put forth so if they are adequate, then ubiquitous semantic phenomena should be simply expressible in set theory, and simple set-theoretic notions should be ubiquitous in semantics. For the most part this seems to be the case—you can do a lot of semantics with membership, subset, intersection, etc.—but obviously it’s not perfect. One point of mismatch is the notion of the Cartesian product (X × Y = {⟨x, y⟩ | xX, yY }) a very straightforward notion in set-theory, but one that does not have a neat analogue in language.

What do I mean by this? Well, consider the set-theoretic statement in (1) and its natural language translation in (2).

(1) P × P ⊆ R

(2) Photographers respect themselves and each other.

What set-theory expresses in a simple statement, language does in a compound one. Or consider (3) and (4) which invert the situation

(3) (P × P) − {⟨p, p⟩ | p ∈ P} ⊆ R

(4) Photographers respect each other.

The natural language expression has gotten simpler at the expense of its set-theoretic translation. This strikes me as a problem.

If natural language semantics is best expressed as set theory (or something similar), why isn’t there a simple bound expression like each-selves with the denotation in (5)?

(5) λX.λY (Y × Y ⊆ X)

What’s more, this doesn’t seem to be a quirk of English. When I first noticed this gap, I asked some native non-English speakers—I got data from Spanish, French (Canadian and Metropolitan), Dutch, Italian, Cantonese, Mandarin, Persian, Italian, Korean, Japanese, Hungarian, Kurdish, Tagalog, Western Armenian, and Russian[2]I’d be happy to get more data if you have it. You can email me, put it in the comments, or fill out this brief questionnaire.—and got fairly consistent results. Occasionally there was ambiguity between plural reflexives and reciprocals—French se, for instance, seemed to be ambiguous—but none of the languages had an each-selves.

My suspicion—i.e. my half-formed hypothesis—is that the “meanings” of reflexives and reciprocals are entirely syntactic. We don’t interpret themselves or each other as expressions of set-theory or whatever. Rather, sentences with reflexives and reciprocals are inherently incomplete, and the particular reflexive or reciprocals tells the hearer how to complete it—themselves says “derive a sentence for each member of the subject where that member is also the object”, while each other says “for each member of the subject, derive a set of sentences where each object is one of the other members of the subject.” Setting aside the fact that this, even to me, proposal is mostly nonsense, it still predicts that there should be an each selves. Perhaps making it sensible, would fix this issue, or vice versa. Or maybe it is just nonsense, but plenty of theories started as nonsense.


1 Yes, I know that there are many other types of model theories put forth
2 I’d be happy to get more data if you have it. You can email me, put it in the comments, or fill out this brief questionnaire.